While the epidemiologists are more concerned with the study design and measures of risk/ association, the biostatisticians are more concerned about the type of variable and the family of tests. Typically, epidemiologists calculate sample size for a given measure of association (Odds Ratio of 2, for example). It is possible for biostatisticans to calculate sample size without having to consider the study design, rather basing the calculation upon the choice of statistical test that will be used during analysis (family of tests mentioned above).

What is the real-world impact of this difference?

If you approach a biostatistican to help with sample size calculation, you will be questioned about the type of variable (primary objective), effect size, and statistical tests for analysis.

However, if you approach an epidemiologist, you will be asked to specify the exact study design, proportions among exposed and unexposed, and the magnitude of Odds Ratio/ Risk Ratio.

Understandably, if you first consult a biostatistician, then an epidemiologist (or vice-versa), you are likely to be very confused. However, on the bright side, doing so will help you obtain clarity on not only the study design, but also the statistical analyses required (both requirements of a study protocol).

Yes, one can obtain different estimates by using different approaches. Even within a single approach, merely choosing a different study design/ family of tests may result in a different value.
This is why the standard practice in the west is to calculate a range of values instead of a single estimate.